I guess the RGB pins of the strip need some kind of voltage, to produce multiple colors instead of just mixing RED, GREEN and BLUE.

No just mixing the R, G B colours at different brightnesses will give you all the colours.

That schematic is no good for what you want because that chip can only switch about 650mA of current in total at any one time with each individual buffer being restricted to a maximum of 500mA.

What should I be using instead? In the store they told me I could use this chip, even though it only provides 500mA for each channel. The strip I'm using uses 1.5A, but only half of the strip will be used, so 0.75A, seperated over 3 channels = 0.25A.

Yes otherwise there is no path to ground for the current, nor common reference for the transistors.

That circuit will work, providing your strip works by grounding the R, G & B signals. But it is a bit of a waste using three arrays only to use one transistor per array.

Might be a waste, but if it works it works These transistors aren't that expensive so I'm all fine with that.

I updated the scheme, you say it should work using the following?

What would happen if I put an analogWrite of 60 on the GREEN-PIN and 160 on the BLUE-PIN, would this create nice blue-ish color a bright Aqua-blue color (Asuming that the transistor doesn't work as a resistor)

Yes the intensity of the colour will change with the PWM value you feed it. By mixing different values you will get different colours.For an example of this see the last half of the video in my project:-http://www.thebox.myzen.co.uk/Hardware/Hexome.html

Take that last circuit. Remove all the black wires that go to ground.Connect pins 1 to 7 together. Do this for all three chips.Connect pins 16 to 10 together. Do this for all three chips.

Then wire pin 8 of all chips to ground. Do not connect pin 7 to anything.

Yes the intensity of the colour will change with the PWM value you feed it. By mixing different values you will get different colours.For an example of this see the last half of the video in my project:-http://www.thebox.myzen.co.uk/Hardware/Hexome.html

Take that last circuit. Remove all the black wires that go to ground.Connect pins 1 to 7 together. Do this for all three chips.Connect pins 16 to 10 together. Do this for all three chips.

Then wire pin 8 of all chips to ground. Do not connect pin 7 to anything.

Ok, so 1-7 are all connected to the Arduino PIN, the 8th goes to GND, 10-16 to the RGB-PIN, but what happens to PIN 9?